Past exam of the medicine course of the University of Oxford / 1867 / First examination / Physics / 3

If a body falls in vacuo, express the velocity, and the distance of the body from the starting point, at any instant in terms of the time which has elapsed since motion commenced.

Account for the comparatively small velocity of a drop of rain when it reaches the earth, and state what is the effective force upon the drop at that time.

Take downward displacement as positive and gravitational acceleration $g$ as constant. In a vacuum, $mdv/dt=mg$, so with initial downward velocity $u$,For release from rest these become $v=gt$ and $s=g{t}^2/2$.

A raindrop encounters an increasing upward drag force as its speed grows. At terminal velocity, gravitational weight is balanced by drag force and the small buoyancy contribution:The drop remains in motion: zero acceleration does not imply zero velocity. In a quadratic drag force model, $F_{\mathrm{drag}}={\rho }_{\mathrm{air}}{C}_{D}A{v}^2/2$, giving $v_t=\sqrt{2(m-{\rho }_{\mathrm{air}}V)g/({\rho }_{\mathrm{air}}{C}_{D}A)}$. That formula assumes a fixed effective area and drag coefficient; sufficiently small drops or deforming large drops need an appropriate model. The question's “effective force” at settled terminal velocity is the approximately zero resultant force, not its nonzero gravitational weight.